Topics
6. Exponential & Logarithmic Functions
topic badge
United States of AmericaVirginia
Algebra 2

6.04 Compare functions across representations

Lesson
Practice

Compare functions across representations

Functions can be represented in a variety of ways, including equations, tables, and graphs. It is important to be able to compare functions whether they are represented in similar or different ways.

Useful information can usually be obtained by comparing characteristics or key features of the functions. Remember that key features include:

  • Domain and range

  • Intercepts

  • Zeros

  • End behavior

  • Relative maximum or minimum value(s)

  • Absolute maximum or minimum value(s)

  • Increasing, decreasing, and constant intervals

  • Asymptote(s)

  • Axis of symmetry

Recall the function families we have studied throughout Algebra 2:

Functions with their general equations and graphs shown below. From left to right, the functions are: Linear, f of x equals x; Quadratic, f of x equals x squared; Square root, f of x equals square root of x; Cubic, f of x equals x cubed; and Cube root, f of x equals cube root of x. Speak to your teacher for more details.
Functions with their general equations and graphs shown below. From left to right, the functions are: Rational-Linear, f of x equals one over x; Rational-Quadratic, f of x equals one over x squared; Exponential, f of x equals b raised to the x, where b is greater than 1; and logarithmic, f of x equals log of x base b, where b is greater than 1; Speak to your teacher for more details.

Examples

Example 1

List the following functions in order from left-most vertical asymptote to right-most vertical asymptote.

  • f(x)=\dfrac{x}{x-3} + 5
  • g(x)=\log{(x-5)} + 6
  • h(x)=\dfrac{x}{x+4} - 3
  • j(x)=\log{(x+2)} - 5
Worked Solution
Create a strategy

To determine the order of the functions based on their vertical asymptotes, we first need to identify the vertical asymptotes for each function.

  • For rational functions, the vertical asymptote occurs where the denominator is equal to zero.
  • For logarithmic functions, the vertical asymptote occurs where the argument of the logarithm is equal to zero.

Once the vertical asymptotes are identified, list the functions in ascending order based on the x-value of the vertical asymptote.

Apply the idea

Determine the vertical asymptotes for each function:

  • For f(x)=\dfrac{x}{x-3} + 5, the vertical asymptote occurs when x-3=0. So, the vertical asymptote is at x=3.

  • For g(x)=\log{(x-5)} + 6, the vertical asymptote occurs when x-5=0. So, the vertical asymptote is at x=5.

  • For h(x)=\dfrac{x}{x+4} - 3, the vertical asymptote occurs when x+4=0. So, the vertical asymptote is at x=-4.

  • For j(x)=\log{(x+2)} - 5, the vertical asymptote occurs when x+2=0. So, the vertical asymptote is at x=-2.

Now, list the functions in order from leftmost to rightmost vertical asymptote:

h(x),\enspace j(x),\enspace f(x),\enspace g(x)

Reflect and check

To check our answer, we can visualize the functions on a graph and observe the order of the vertical asymptotes. The graph should show that the vertical asymptotes are ordered from left to right as follows: h(x) at x=-4, j(x) at x=-2, f(x) at x=3, and g(x) at x=5. This confirms our answer.

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
y

Example 2

Consider the following square root and logarithmic functions:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
a

Compare the zeros of the functions.

Worked Solution
Create a strategy

The zero of a function occurs at y=0, which is where the graph crosses the x-axis. We can label the x-intercepts on each graph to more clearly see where they are.

Apply the idea
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Looking at the graphs, we can see that the zero of both functions is at the point \left(3, 0\right).

Reflect and check

Had we known the equation for each graph, we would have been able to algebraically confirm our answer by substituting 3 in for x in each equation and ensure each results in a value of 0.

b

Compare the domain and range.

Worked Solution
Create a strategy

We will analyze the graphs of the functions to determine their domain and range. The domain is the set of all possible x-values, while the range is the set of all possible y-values that the functions can take.

Apply the idea

Observing the graph of the square root function, we can see that it starts at x=-1 and extends to the right infinitely. This means the domain of the function is x \geq -1 or [-1,\infty). Observing the y-values, the graph starts at the point (-1, -2) and increases without bound. Thus, the range of the function is y \geq -2 or [-2,\infty).

Observing the graph of the logarithmic function, we can see that it starts at x=2 and extends to the right infinitely. This means the domain of the function is x \gt 2 or (2,\infty). Observing the y-values, the graph extends downward without bound and increases without bound. Thus, the range of the logarithmic function is all real numbers, denoted as (-\infty,\infty).

Reflect and check

We can also write the domain and range of the two functions in set notation:

For the square root function:

  • Domain: \{x|x\geq -1\}

  • Range: \{y|y\geq -2\}

For the logarithmic function:

  • Domain: x > 2

  • Range: \{y|y\in \Reals\}

c

Compare the absolute maxima or minima, if any.

Worked Solution
Create a strategy

Finding the absolute maxima or minima involves identifying the highest or lowest points of the function. These will correspond with the endpoints of the function's range. Since we have already found the domain and range, we can use this information to identify any absolute extrema.

Apply the idea
  • Domain of square root function: [-1,\infty)
  • Range of square root function: [-2,\infty)

  • Domain of logarithmic function: (-\infty,\infty)

  • Range of logarithmic function: (-\infty,\infty)

From the domain and range information, we can see that the square root function starts at the point (-1, -2) and increases indefinitely. As the function continuously increases, there is no absolute maximum. The absolute minimum occurs at the point (-1, -2).

For the logarithmic function g(x)=\log_{10}(x-2), we observe from the range that the function decreases and increases without bound. This implies that there is no absolute maximum or minimum.

Reflect and check

By analyzing the graphs of the square root function and the logarithmic function on the same Cartesian plane, we can visually identify any absolute maxima or minima. The square root function has an absolute minimum at the point (-1, -2), while the logarithmic function has neither an absolute maximum nor an absolute minimum, which confirms our answer.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d

Compare the end behavior.

Worked Solution
Create a strategy

Observe the graphs of the functions to determine how the functions behave as x decreases and increases. Focus on the direction and behavior of the functions as they move to the left and to the right on the graph.

Apply the idea

Observing the graph of the square root function, we can see that as x approaches -1, the function value approaches -2. As x approaches positive infinity, the function f(x) increases slowly without bound, so the function approaches positive infinity.

For the logarithmic function, we can see that as x approaches 2, the function approaches negative infinity. As x approaches positive infinity, the function g(x) increases slowly without bound, so the function approaches positive infinity.

The end behavior can be written as follows

Square root function:

as x\to -1, f(x)\to -2 and as x\to \infty, f(x)\to \infty

Logarithmic function:

as x\to 2, f(x)\to -\infty and as x\to \infty, f(x)\to \infty

We can see similarities amongst the functions' end behavior. For both functions, as x decreases, both functions are restricted, so x does not decrease infinitely. Additionally, as x approaches positive infinity, both functions approach positive infinity.

Reflect and check

Remember that when we look at increasing or decreasing intervals of functions, we usually consider how the function values change as we move from left to right.

When we are looking at the end behavior of functions, we consider how the function values change as the input values get smaller or larger instead.

Example 3

Consider the function f(x) = -(x+2)^3 and the function g(x) represented by the table of values.

x-1012345
g(x)-27-8-101827
a

Compare and contrast the intervals over which the functions are increasing or decreasing.

Worked Solution
Create a strategy

To compare and contrast the intervals of increasing or decreasing for the functions, we must first determine where, and if, the graph is increasing or decreasing. Since f\left(x\right) is a cubic function, we know it will either increase everywhere or decrease everywhere.

For g\left(x\right), we will examine the pattern in the outputs.

Apply the idea

For the function f(x)=-(x+2)^3, it is a cubic function whose graph has been reflected over the x-axis and translated left 2 units compared to the parent function. The parent function, y=x^3, is increasing everywhere, so the reflection will cause our graph to flip. This means f(x) is decreasing for (-\infty, \infty).

For the limited domain given for g(x), we can see that as the x-values increase, the y-values are also increasing. In addition, the function appears to have a cubic relationship. This implies g(x) is increasing everyhwere.

b

Identify and compare the zeros.

Worked Solution
Create a strategy

We can find the zero of f(x) algebraically by setting the equation equal to zero and solving for x. To find the zero for g(x), we will use the table of values.

Apply the idea

Let's first find the zero of the function f(x):

\displaystyle -(x+2)^3\displaystyle =\displaystyle 0Set the function equal to 0
\displaystyle (x+2)^3\displaystyle =\displaystyle 0Divide both sides by -1
\displaystyle x+2\displaystyle =\displaystyle 0Cube root both sides
\displaystyle x\displaystyle =\displaystyle -2Subtract 2 from both sides

To find the zero of the function g(x), we can use our table of values. We know the zero of a function occurs where the y-value of the function equals 0. We can see from the table that the function has a zero at x=2.

We can see that both f(x) and g(x) have exactly one zero. The zeros of the two functions are similar, but one is 2 units to the left of the origin while the other is 2 units to the right of the origin.

Example 4

The graph of function f \left(x \right) shown follows the vertical height of Ameth's golf ball after being hit into the air and falling on the fairway:

Ameth's golf ball
1
2
3
4
5
6
7
8
9
10
11
\text{Time (seconds), } x
5
10
15
20
25
30
\text{Height (feet), } f \left(x \right)

The equation of function g(x) shown follows the vertical height of Massiel's golf ball. She did not get a good hit on the ball and, after a few seconds in the air, the ball hit a tree and ricocheted off into a sand trap where it got stuck.

g(x) = \begin{cases} 6 \sqrt[3] x, & 0 \leq x < 4 \\ -2.4x+19.125 , & 4 \leq x < 7.5 \\ \dfrac{1}{2} (x-9)^2, & 7.5 \leq x < 9 \end{cases}

a

Compare the intervals where each golf ball is increasing in height.

Worked Solution
Create a strategy

We can use the graph to determine when f(x) is increasing in height. For g(x), we will examine the functions in the piecewise function to determine when it is increasing.

Apply the idea

Looking at the graph of f(x), we can see the function is increasing in height until it reaches its maximum at x=5, and then is decreasing for rest of graph.

f(x) is increasing in height over the interval (0,5).

For the function g(x), we must take a look at what each piece of the piecewise function is representing. The function starts with a cube root function, which is increasing everywhere. Then at x=4, it becomes a linear function with a negative slope, which is decreasing. The final piece is an upward facing parabola, which is decreasing until the vertex at x=9.

g(x) is increasing in height over the interval (0,4).

b

Describe what g(x)=-2.4x+19.125 for 4\leq x\lt 7.5 represents in context.

Worked Solution
Create a strategy

Graph g \left( x \right), then use the context to interpret the piecewise function graphically.

Apply the idea
Massiel's golf ball
1
2
3
4
5
6
7
8
9
10
11
\text{Time (seconds), } x
1
2
3
4
5
6
7
8
9
10
11
\text{Height (feet), } f \left(x \right)

In this part of the function, the golf ball's height decreases linearly with time. This occurs within the time interval 4 to 7.5 seconds. During this time, the ball hits a tree and ricochets off, causing it to lose height as it moves towards the sand trap.

c

Determine whose ball goes higher.

Worked Solution
Create a strategy

Compare the maximum values of the graphs of Ameth's ball and Massiel's ball.

Ameth's golf ball
1
2
3
4
5
6
7
8
9
10
11
\text{Time (seconds), } x
5
10
15
20
25
30
\text{Height (feet), } f \left(x \right)
Massiel's golf ball
1
2
3
4
5
6
7
8
9
10
11
\text{Time (seconds), } x
1
2
3
4
5
6
7
8
9
10
11
\text{Height (feet), } f \left(x \right)
Apply the idea

Ameth's ball reaches a maximum height of 30 \text{ ft} while Massiel's ball reaches a maximum height of just over 9 \text{ ft}. Ameth's ball clearly goes higher.

Reflect and check

We used the graph of g(x) to approximate the maximum height, but we could have found the precise height by substituting x=4 into the function g(x)=-2.4x+19.125.

\displaystyle g(4)\displaystyle =\displaystyle -2.4(4)+19.125Substitue x=4
\displaystyle {}\displaystyle =\displaystyle -9.6 +19.125Multiply
\displaystyle {}\displaystyle =\displaystyle 9.525Add

Massiel's ball reaches a maximum height of 9.525\text{ ft}.

Idea summary

Real-world contexts can be modeled by various types of functions. We can compare different types of models for contextual situations using their key features:

  • Domain and range

  • Intercepts

  • Zeros

  • End behavior

  • Relative maximum or minimum value(s)

  • Absolute maximum or minimum value(s)

  • Increasing, decreasing, and constant intervals

  • Asymptote(s)

  • Axis of symmetry

Outcomes

A2.F.1

The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.F.1a

Distinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.

A2.F.1e

Compare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.

A2.F.2

The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

A2.F.2a

Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.

A2.F.2b

Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.

A2.F.2c

Determine the intervals on which the graph of a function is increasing, decreasing, or constant.

A2.F.2d

Determine the location and value of absolute (global) maxima and absolute (global) minima of a function.

A2.F.2e

Determine the location and value of relative (local) maxima or relative (local) minima of a function.

A2.F.2g

Describe the end behavior of a function.

A2.F.2h

Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).

What is Mathspace

About Mathspace